Download Technical Note Transducers in Fluid Flow Applications

Transducers in Fluid Flow Applications
Technical Note
INTRODUCTION
People have been building fluid systems for many
millennia and the knowledge of how to measure
pressure in the fluids is centuries old. So what’s so
special about the line of transducers introduced by
Honeywell? In a nutshell, the solid state transducer
allows much better sensing and, therefore, much better
control of fluid systems for a given amount of money
spent on system control.
To give the electrical engineer a better understanding
of the strange worlds of mechanical, chemical,
aeronautical, civil, and acoustical engineers, we will
describe the three basic classes of transducer
applications. All transducer usages will fit into one of
the following categories:
this broad view, on a calm day the world is a pressure
vessel. Likewise, a vacuum chamber is a pressure
vessel. As indicated in Figure 1, the sensor can be
plumbed into the pressure vessel, thus adding its own
package volume and that of the plumbing to the volume
of the pressure vessel. Where the pressure vessel is
small, this additional volume must be considered if the
measurement or control function involved seeks some
relationship between pressure and volume.* It is
important to recognize that the change in volume of the
transducer package with pressure change is entirely
negligible.
FIGURE 1
1. Pressure Vessel
2. Open Flow
3. Closed Flow
Each is thoroughly described and the key equations of
each class are derived.
For a mental model think of “open flow” as represented
by the flow of a fluid in which the main energy involved
is simply kinetic. Water in an aqueduct or a long pipe
where the potential energy at the dam up in the
mountains or the compressive energy of the pump, has
long since been converted to the kinetic energy of the
flowing stream.
As indicated in Figure 2, the sensor can be totally
enclosed within the pressure vessel, working fluid
chemistry allowing, so that little volume change in the
system is experienced. When the volume of the vessel
is large compared to that of the sensor package, as is
usually the case, the two alternative hook-ups are
equivalent.
FIGURE 2
For “closed flow,” think of a compressor as used for
example in refrigeration systems. Here the fluid density
is changing and thus significantly contributing to the
energy of the system, still primarily kinetic.
In a “pressure vessel” a stationary fluid is assumed.
Now on to the system modeling.
“PRESSURE VESSEL” APPLICATION
The simplest application of an absolute pressure
transducer is the direct measurement of absolute
pressure of a fluid at rest within a pressure vessel.
The concept of a pressure vessel is very general
here ... it merely represents that container of fluid
which keeps the fluid free of dynamics. Thus, using
Sensing and Control
Finally, when the walls of the pressure vessel
disappear, we have a barometer ... a very good
application of Honeywell absolute sensors.
*Care must be taken to avoid any temperature
gradients between the pressure vessel and the tranducer.
Technical Note
Transducers in Fluid Flow Applications
Pressure — Temperature Measurements
Another important family of “pressure vessel”
applications involves the measurement of temperature
used in conjunction with an absolute pressure reading.
The integrated temperature sensing diode reads the
temperature of the silicon diaphragm in the transducer
package. This temperature signal may be used for
the additional temperature compensation of the
pressure signal.
The accuracy of the transducer depends primarily upon
the temperature coefficients of the pressure signal. This
is typically one to two orders of magnitude looser than
the precision, which depends upon hysteresis and
deadband. Therefore, for such devices, if one were to
calibrate the pressure signal temperature error in terms
of the temperature signal, then a correction table could
be established to increase its accuracy to the same
order as its precision.
Other Fluid Functions
By suitably combining the temperature with the
pressure signal, other fluid variables within the
pressure vessel can be formulated. Two important fluid
variables are density and heat content.
Density of a known gaseous fluid can be calculated
without knowledge of the fluidic thermodynamic
process going on simply by measuring simultaneously
the pressure and temperature at the same point as
described in equation 1.
Equation 1:
Equation 2:
Where dh = change in heat content
C = inverse Joule’s constant
3
If one were willing to track the fractional change of
temperature and density as well as that of pressure,
then complete knowledge of the state of the enclosed
fluid as well as a complete characterization of the
changing process results. Expressing equation 1 in
terms of fractional changes yields equation 3.
Equation 3:
where
Define a process characterization constant C4 as
shown in equation 4 and substitute in equation 3.
Equation 4:
where
p = Density
P = pressure
T = temperature
C = inverse gas contsant
2
Knowing the pressure, temperature and the density of
the fluid at one or more points simultaneously, much
can be learned about the nature of the processes
causing change in the fluid. In a bounded pressure
vessel, one variable often of concern is the heat
transfer or energy exchange within some portion of a
cycle or during some specific period. The change in
heat content at a specific point in the system can be
traced most easily in the bounded pressure vessel by
tracking the fractional change in pressure as shown in
equation 2.
2 Honeywell • Sensing and Control
Figure 3 is a table of this characterization constant
under a number of important practical conditions.
So, by measuring absolute pressure changes and
absolute temperature changes in an enclosed fluid
system, one is able to characterize the process which
caused the change of fluid properties. The process
characterization constant (C4) relates original oil
changes in system energy to changes in fluid state
defined by pressure, temperature, and density.
Technical Note
Transducers in Fluid Flow Applications
FIGURE 3
State and Process Characterization
Note: In all cases dP dT, dp represent changes with time of the absolute values of P, T, and p.
Phase Change
The process characterization constant poorly defines
thermodynamic processes where changes of phase
are involved. Where a phase change from gaseous to
liquid fluid is involved, a characteristic energy transfer
called latent heat (HLATENT) occurs in accordance with
expression given by Clapeyron, shown as equation 5.
Equation 5:
Generally, measurements of pressure and temperature
in a multiphase system are difficult. Often for the
purpose of measurement, a small parallel tap is made
into the main flow channel so as to divert a negligible
portion of the fluid flowing; then a sudden expansion is
performed in the tap, and a small tap is led back into
the main flow channel. This type of device is called a
flash chamber and is shown in Figure 4.
FIGURE 4
where
C2 = inverse gas constant
For various fluids entering phase change, a new
process characterization constant (C5) given by
Clausius/Clapeyron, shown in equation 6.
The flash chamber accomplishes the conversion of
multiphase system (liquid and gas) to a single phase
system (gas).
Equation 6:
Such a chamber is simply a pressure vessel
susceptible to analysis by the aforementioned modeling
techniques.
or
Thus, we have developed the equations of state for
fluids undergoing thermodynamic processes with and
without phase changes so as to characterize these
processes via measurement of pressure and
temperature within a pressure vessel using the
absolute transducer.
OPEN FLOW APPLICATIONS
Just as pressure vessel processes were characterized
by thermodynamic changes in fluid media rather than
kinetic or spatial changes, open flow conditions and
processes are best characterized as those in which we
are primarily concerned with the kinetic properties of a
fluid system and their changes, rather than the state
and changes of state of a controlled volume of fluid.
Whereas in the pressure vessel measurements involve
absolute pressure, the kinetic state of an open flow
system most often requires gage and differential
pressure transducers.
Honeywell • Sensing and Control 3
Technical Note
Transducers in Fluid Flow Applications
The form of Bernoulli’s flow equations most applicable to
open flow conditions is given in equation 7.
Equation 7:
kinetic flow head, and its potential flow head remains
constant. In most open flow situations the variable of
major interest is flow velocity. In situations where it is
convenient to measure both stagnation pressure and
static pressure at essentially the same point
(y is constant) a differential pressure transducer can be
used in a method similar to that of a pilot-tube as shown
in Figure 5. Equation 7 can then be rearranged as
shown in equation 8.
FIGURE 5
Flow Rate Pressure Square Law Relationship
Equation 8:
y = Potential flow head
This particular form of Bernoulli’s equation is valid
where fluid density remains reasonably constant. As
seen in the definition of terms (equation 7), the concept
of a head is key to open flow applications. Easiest to
conceive is the potential head, wherein a particle of fluid
at a different height than at some other location has
some potential to flow to that other location. That
potential is proportional to the height difference. It is
small wonder, therefore, that dimensions of a head are
those of a column of fluid; inches of mercury, feet of
water, millimeters of mercury or torr. If that particle of
fluid were to fall, so as to change location or flow,
without changing pressure within that particle, then that
potential flow head would convert to a kinetic flow head.
Another useful concept in equation 7 is that of
stagnation. If you were waiting with a catcher’s mitt at a
fixed location and a particle of fluid at a given pressure
flowed into the mitt, in order to stop that particle with
your mitt you would have to bring it to rest. In doing so,
assuming the fluid particle remained at the same
density and temperature, the particle’s pressure head
would increase by the absorbed kinetic head. If you
were then to pull the mitt down to ground level, the
stagnation head would in addition increase by the
potential head. Therefore, the stagnation head of a
given fluid particle is an expression of that particle’s
pressure were it suddenly brought to rest and dragged
to the bottom. Bernoulli’s equation indicates that however that particle may meander under open flow
conditions; the sum of its static pressure head, its
4 Honeywell • Sensing and Control
Applications where this type of measurement is
common are air speed indication for airplanes and
weather balloons, water speed for boats and
aqueducts and irrigation ditches, ventilation control,
sewage processing, industrial mixing, and others. More
generally, where y varies in open flow conditions,
changes in flow conditions due to geometric changes
in the flow boundaries are monitored by
measurements of fluid level changes and fluid
pressure changes. Equation 9 gives the applicable
form of Bernoulli’s equation.
Equation 9:
where dv, dp, dy are changes with distance
Figure 6 shows this general case of open flow
boundary condition monitoring. Notice that both the
fluid level and each of the gage transducers used are
vented to atmosphere for reference. In this particular
case, the use of a differential transducer is not
practical due to the physical separation of the points
of interest.
Technical Note
Transducers in Fluid Flow Applications
Two transducers must be used. Examples of
applications include weir and dam flow and level control,
oceanographic measurements, and for the special
cases of near zero velocities, general level control.
FIGURE 6
Flowmeter Output with Exaggerated Pressure Transducer
Common- Mode Inaccuracy
ACOUSTICS
One special form of open flow application is the case
where the pressure wave velocity is high compared to
the fluid flow velocity. Such applications are called
acoustic. The acoustic energy of a sound wave is
directly proportional to the sound pressure as shown in
equation 10.
Equation 10:
The schemes of both Figures 5 and 6 are ideal for
accuracy improvement via auto-referencing
Once again there are some cautions that must be
respected in the application of transducers in open flow
conditions. Perhaps the most often overlooked and
frustrating caution involves temperature gradient. As
was explained in the pressure vessel section, although
good transducers are temperature compensated,
temperature gradients within the transducer are deadly
enemies producing errors that defy all compensation
techniques. If the temperature signal from the
transducer is used in determining the density of the
working fluid, then the transducer wants to be thermally
well-coupled. However, if the atmospheric
environmental temperature is significantly different from
that of the working fluid, a severe gradient of
temperature can occur within the transducer unless the
transducer is totally immersed in the working fluid
(a condition usually impractical). Fortunately, in most
open flow applications, the temperature reading is not
needed to determine density and the dynamics of the
flow do not prohibit fairly long tube lengths from the
transducer to the point of measurement. The result is
that it is practical to provide a situation wherein the
working fluid within the transducer is at the same
temperature as the surrounding environment.
The IC pressure transducers are ideally suited for
coupling to VCOs for FM transmission (see Signal
Conditioning section) the method favored by all modern
trends in instrumentation. Among the many advantages
are extremely low noise susceptibility even when
bundled with many other signal carrying cables, ease of
conversion to digital signals to facilitate interface with
modern control logic or input to general purpose digital
computers, and ease of combination with other analog
FM signals to form a composite system variable.
For acoustics, the open flow approximation of an
“incompressible, isothermic medium” must be replaced
by the model of a “stationary, isentropic medium.” That
is, the propagating pressure wave shakes each fluid
particle about its stationary location proportional to the
sound pressure constituting a dynamic change in
density (dp). As shown in equation 11, the
proportionality constant between the change in the fluid
density and the sound pressure is the inverse squared
speed of sound in that medium.
Equation 11:
where “a” is the speed of sound.
Sound pressure measurements should be made with
differential transducers in which the reference point
monitors the quiescent pressure level.
High frequency response can be severely limited by
bounded chambers. In fact, just the tube and lid can
limit response normally in the tens of kilo Hertz to the
low kilo Hertz region. However, the silicon pressure
diaphragm response alone with the tube and lid
removed or when liquid coupled is such that calibrated
sound pressure measurements can be achieved at
frequencies well above 50 kilo Hertz. Thus, pressure
transducers are capable of calibrated pressure measurements in open flow conditions ranging in dynamics
from static to ultrasonic frequency.
CLOSED FLOW APPLICATIONS
Pressure vessel applications require the investigation of
time dependent variables at one point. Open flow
applications require the investigation of time
independent variables at many points. In closed flow
applications, we need to investigate both kinds of
variables. That is, the processes at work within the fluid
Honeywell • Sensing and Control 5
Technical Note
Transducers in Fluid Flow Applications
system involve state change energies of like order of
magnitude as kinetic energies. Perhaps the most
obvious examples of systems reliant on such processes
are engines of all kinds, as well as refrigerators and air
conditioners, compressors, gas pipelines, fire
extinguishers, gun cartridges and explosives.
In closed flow, the equation used to model the flow
between two locations must be altered to include the
change of state.
Again, rather than the incompressible and isothermic
conditions applicable to liquids in open flow, heat
transfer and in fact heat loss due to flow conditions is
experienced. Equation 15 presents Bernoulli’s equation
for this situation with the energy loss due to obstruction
indicated in terms of change in heat content. For
extremely long, thin, rough-walled, closed-flow vessels,
the fractional loss can actually approach unity.
Equation 15:
Equation 12:
where
In equation 12, hSTAGNATlON represents the total equivalent
heat energy of the system and is constant; h is the heat
content of a fluid particle at a particular location such
that (hSTAG - h) represents the kinetic and potential flow
energy. In open flow, the density of the fluid is assumed
to be known and constant. Consider a working fluid that
is gaseous and whose density variation may be
described by equation 1 of the pressure vessel
applications section. Equation 13 shows the substitution
of equation 1 in equation 12.
Equation 13:
Structures are used that tend to minimize the distance
between sensing points as well as minimize the
fractional loss of energy. Commonly used structures
range from orifices where losses are up to half of the
total energy and sensing points are very closely
spaced, to Venturi’s where losses are negligible but
sensing points are widely spaced. Equation 16, derived
from equation 15 shows a typical calibration equation
for a specific obstruction over a specific range of flow.
Equation 16:
In the same manner in which equation 8 was derived
from equation 7 in the section on open flow
applications, equation 14 is derived from equation 13 to
show flow velocity.
Equation 14:
Thus, where the system allows a Pitot tube
measurement of the two absolute pressures and
temperatures at one point, the closed flow can be
modeled by equation 14 and two transducers. Common
applications include high velocity air flow as in
compressors, and engine manifold gas flow.
To measure closed flow in small flow channels, it is
usual to insert an obstruction in the passage between
two points at which fluid properties can be sensed.
6 Honeywell • Sensing and Control
Figure 7 shows some typical flow measurement
obstructions.
FIGURE 7
Transducers in Fluid Flow Applications
Technical Note
It must be remembered that these are rule of thumb
examples and fluids undergoing drastic thermodynamic
changes will not approximate these processes for these
flow conditions. For example, the nearly lossless Venturi
of streamlined flow conditions forms a roaring energy
loser when turbulent hot gases are pushed through so
as to become the converging-diverging nozzle of
rocketry fame.
of two absolute pressure transducers along with two
temperature transducers (to achieve a more accurate
modeling equation) too sophisticated. The tradition of
simple single variable models using few highly accurate
transducers to achieve moderate system accuracy is so
well founded in the measurement and control industries
that until recently only manufacturers of low volume,
high accuracy, high cost transducers existed.
OPEN VS CLOSED FLOW
Conventionally, differential pressure transducers rather
than absolute transducers are used for flow
measurement. Three arguments are popular for
maintaining this convention, one of them very good and
the other two rather weak.
In most closed flow applications the variables of interest
are functions of absolute pressure, differential pressure,
and temperature, with respect to time and location. If
the system modeler desires a certain tolerance on this
complex function he may either gather single variable
data at multiple locations or multivariable data at few
locations.
In low flow rate applications the dynamic head is most
often small compared to the static head. The
incompressible isothermic model, expressed in the
equations of the open flow applications section, is often
appropriate under such low flow conditions. The
resulting simple expression of Bernoulli’s equation that
leads to the use of differential pressure transducer is
given by equation 17.
In either case, auto-referencing techniques can greatly
improve system accuracy.
Equation 17:
In addition to being the analytically obvious best choice
when equation 17 is an appropriate model of flow
conditions, a single differential transducer can be made
to perform with much greater accuracy than a pair of
gage or absolute transducers of comparable quality
because the differential transducer need only range
both the dynamic and static heads. Since the static
head is not called for in equation 17, the added ranging
requirement merely adds common mode error to the
measurement. Under these conditions the differential
pressure transducer convention for flow measurement
is justified and the differential transducers are highly
recommended, particularly with auto-referencing.
Equation 18:
Traditionally, even in cases where more complex
Bernoulli models (equation 18) should be used, the
simple incompressible model is substituted. One
argument claims that one can always make up for
errors in the modeling equation by specifying greater
accuracy in the differential pressure transducer.
Proponents of this argument consider the alternate use
Honeywell • Sensing and Control 7
Transducers in Fluid Flow Applications
WARRANTY/REMEDY
Honeywell warrants goods of its manufacture as
being free of defective materials and faulty workmanship. Contact your local sales office for warranty
information. If warranted goods are returned to
Honeywell during the period of coverage, Honeywell
will repair or replace without charge those items it
finds defective. The foregoing is Buyer’s sole
remedy and is in lieu of all other warranties,
expressed or implied, including those of merchantability and fitness for a particular purpose.
Specifications may change without notice. The
information we supply is believed to be accurate and
reliable as of this printing. However, we assume no
responsibility for its use.
Sensing and Control
www.honeywell.com/sensing
Honeywell
11 West Spring Street
Freeport, Illinois 61032
008113-1-EN IL50 GLO 1104 Printed in USA
Copyright 2004 Honeywell International Inc.
All Rights Reserved
Technical Note
While we provide application assistance personally,
through our literature and the Honeywell web site, it is
up to the customer to determine the suitability of the
product in the application.
For application assistance, current specifications, or
name of the nearest Authorized Distributor, check the
Honeywell web site or call:
1-800-537-6945 USA/Canada
1-815-235-6847 International
FAX
1-815-235-6545 USA
INTERNET
www.honeywell.com/sensing
[email protected]